Below are listed the major Definitions, Theorems and Propositions related to Groups, Subgroups, and Cyclic Groups

1. is a group under addition modulo n. With identity 0 and the inverse of is the

2. The number of elements of a group is its order

3. The order of an element g, which is denoted by , in a group G is the smallest positive integer n such that

4. Cyclic group and g is called a generator of

5. The order of the generator is the same as the order of the group it generated.

6. Let G be a group, and let x be any element of G. Then, is a subgroup of G.

7. Let be a cyclic group of order n.

Then if and only if gcd(n,k)=1.

8. Every subgroup of a cyclic group is cyclic.

9. Every cyclic group is Abelian.

10. If G is isomorphic to H then G is Abelian if and only if H is Abelian.

11. If G is isomorphic to H then G is Cyclic if and only if H is Cyclic.

12. Lagrange’s Theorem: If G is a finite group and H is a subgroup of G, then the order of H divides G.

13. In a finite group, the order of each element of the group divides the prder of the group.

14. A Group of prime order is cyclic.

15. For any where e is the identity element of the group G.

16. An infinite cyclic group is isomorphic to the additive group

17. A cyclic group of order n is isomorphic to the additive group of integers modulo n.

18. Let p be a prime. Up to isomorphism, there is exactly one group of order p.

19. Let (G, *) be a group and . If , then , where is the identity element of the group G.

20. In a Cayley table for a group (G, *), each element appears exactly once in each row and exactly once in each column.

21. A group (G,*) is called a finite group if G has only a finite number of elements. The order of the group is the number of its elements.

22. A group with an infinite number of elements is called an infinite group.

23. If (G,*) is a group, then ({e},*) and (G,*) are subgroups of (G,*) and are called trivial.

24. Let G be a group and H be a non-empty subset of G. then H is a subgroup of G if and only if for all .

25. Let G be a group and H be a finite non-empty subset of G. then H is a subgroup of G if and only if for all .

26. Let be a finite group of order n.

Then

27. Let be a finite cyclic group. Then the order of g equals the order of the group.

28. A finite group G is a cyclic group if and only if there exists an element such that the order of this element equals the order of the group ().

29. Let G be a finite cyclic group of order n. Then froe every positive divisor d of n, there exists a unique subgroup of G of order d.

30. Let G be a group of finite order n. Then the order of any element x of G divides n and

31.Let G be a group of prime order. Then g is cyclic.

Hope these help!!

Statistics: Posted by miranda — Fri Mar 29, 2013 8:26 pm

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Can anyone help me with the fundamental theorems and propositions on Group Theory??

Thanks

Statistics: Posted by elizabeth — Fri Mar 29, 2013 8:19 pm

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The definition of isomorphism is

Two groups {G,*} and {H,o} are isomorphic if:

- there is a bijection

and for every

and for your question we have

is a group under addition

and is a group under multiplication, and is a bijection from into

defined by is an

isomorphism between and

Since f has inverse function and we have the following

Hope these help

Statistics: Posted by miranda — Thu Mar 07, 2013 8:13 pm

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The basic principles about Abelian Groups are:

First of all I remind you the definition of Group (G,*)

Let G be a non-empty set on which a binary operation * is defined . We say G is a Group under this operation if each of the following four properties hold:

1. G is closed under * , i.e., for every

2. The operation * is associative on G,

i.e., for every

3. The operation * has an identity element in G,

i.e., for every

4. Each element of G has an inverse under *,

i.e., for every

A group {G,*} is called Abelian if for every

Now concerning your question

First we must construct the Cayley table in order to determine closure and the existence of identity and inverse elements.

the set under the addition modulo 3

Closure: The addition modulo 3 on the set is closed

i.e. for every

Associative: The addition on is Associative

i.e. x+(y+z)=(x+y)+z for every

Identity: For every element there is an element, the zero, which is the identity element in under addition.

i.e. x+0=0+x=0

Inverse: The identity element {0} appears once in every row and every column, so every element has an iverse.

Commutative: The addition on is Commutative

i.e. x+y=y+x for every

As for the order you have to know the following

The order of a group {G,*} is the number of elements in G

The order of an element x of a group {G,*} is the smallest positive integer n for which with e the identity element.

A group {G,*} is said to be finite if it has a finite order and is said to be infinite if it has infinite elements.

So in you exercise the identity element has order 1 since 0=0

the element 1 has order 3 since 1+1+1=0

and finally the element 2 has also order 3 since 2+2+2=0

Hope these help

Statistics: Posted by miranda — Thu Mar 07, 2013 8:07 pm

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The basic principles about Abelian Groups are:

First of all I remind you the definition of Group (G,*)

Let G be a non-empty set on which a binary operation * is defined . We say G is a Group under this operation if each of the following four properties hold:

1. G is closed under * , i.e., for every

2. The operation * is associative on G,

i.e., for every

3. The operation * has an identity element in G,

i.e., for every

4. Each element of G has an inverse under *,

i.e., for every

A group {G,*} is called Abelian if for every

Also Given a group {G,*} we have the following properties:

Left cancellation law: If x*y=x*z then y=z

Right cancellation law: If y*x=z*x then y=z

Now concerning your question,

The set under the addition is an Abelian group since

Closure: The addition on is closed

i.e. for every

Associative: The addition on is Associative

i.e. x+(y+z)=(x+y)+z for every

Identity: For every element there is an element, the zero, which is the identity element in under addition.

i.e. x+0=0+x=0

Inverse: For every element there is an element, which is its opposite in .

i.e. x+(-x)=(-x)+x=0

Commutative: The addition on is Commutative

i.e. x+y=y+x for every

Hope these help

Statistics: Posted by miranda — Thu Mar 07, 2013 7:59 pm

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The basic principles about iverse element are:

- An element in a set A is an inverse element for an operation * defined over A if

for every element

where e is the identity element.

- For an associative operation *, an element x admits a left-inverse x’and a right-inverse x’’, then these two identities are equal.

- For an operation * on a set A having an identity element then every invertible element admits a unique inverse.

Concerning your question,

First we are going to find the identity element as follows

suppose that y be a right-inverse element of x,

then

The right-inverse element y is also a left inverse of x since

Therefore every element has an inverse of the form

Statistics: Posted by miranda — Thu Mar 07, 2013 7:55 pm

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The basic principles about identity element are:

- An element e in a set A is an identity element for an operation * defined over A if

e*x=x*e=x for every element

- If an operation * admits a left-identity and a right-identity , then these two identities are equal.

- If a binary operation * on a set A admits an identity element e, then this element is unique.

Concerning your question, suppose that the identity element is e, then

which is not unique, since is depending on x. So this operation has no identity.

Hope these help

Statistics: Posted by miranda — Thu Mar 07, 2013 7:53 pm

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Since x*y=xy+4 and y*x=yx+4= xy+4= x*y, the operation is commutative.

(x*y)*z=(xy+4)*z=(xy+4) z+4=xyz+4z+4

x*(y*z)=x*(yz+4)=x (yz+4)+4=xyz+4x+4 which is different than (x*y)*z

thus the binary operation * is not associative.

Statistics: Posted by miranda — Thu Mar 07, 2013 7:51 pm

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Since x*y=x+y-4 and y*x=y+x-4= x+y-4= x*y, the operation is commutative.

(x*y)*z=(x+y-4)*z=(x+y-4)+z-4=x+y+z-8

x*(y*z)=x*(y+z-4)=x+(y+z-4)-4=x+y+z-8

thus the binary operation * is associative.

Statistics: Posted by miranda — Thu Mar 07, 2013 7:50 pm

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How can we show that the function defined by is an isomorphism between and

Thanks

Statistics: Posted by elizabeth — Thu Mar 07, 2013 7:47 pm

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How can we show that the set under the addition modulo 3 is an Abelian group? and what is the order of each element?

Thanks

Statistics: Posted by elizabeth — Thu Mar 07, 2013 7:46 pm

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How can we show that the set under the addition is an Abelian group?

Thanks

Statistics: Posted by elizabeth — Thu Mar 07, 2013 7:45 pm

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Let the binary operation on R defined be x*y=x+y+24. Does each element have an inverse element?

Thanks

Statistics: Posted by elizabeth — Thu Mar 07, 2013 7:45 pm

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Has the binary operation on R defined be x*y=xy+4 an identity element?

Thanks

Statistics: Posted by elizabeth — Thu Mar 07, 2013 7:39 pm

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How can we show if the following binary operation on R defined be x*y=xy+4 is commutative or associative.

Thanks

Statistics: Posted by elizabeth — Thu Mar 07, 2013 7:38 pm

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